1) [30 points] The Lagrange density for a two component self-interacting scalar field is C = [(@ $1)(2"$1) + (2,$2)(2" $2)] + ,Maldi+ $31- -103+$312. (1) For uq > 0 with the sign conventions as shown, the vacuum has broken symme- try and there is a Goldstone boson. (a) [15] A nice way to see this is to consider the change of variables to two new fields p(x, t) and x(x, t) with pi(x, t) = p(x, t) cos x(x, t) and $2(x, t) = p(x, t) sin x(x, t) or o(x, t) = 1(x, t) + ip2(x, t) = p(x, t)ex(I,t) . Show that p is the massive field and x is the Goldstone boson. Compute the mass of the Higgs (rho(x, t) = po + n(x, t)) field to make sure you get the same answer as in class. (b) [10 points] Perhaps you recall that Eq. 1 has a conserved current due to the symmetry sin a - sin a cosa ) ( m ). (2) Re-derive the expression for the associated Noether current in terms of ($1, $2). Then make the change of variables into (p, x) and reconstruct J". Notice (and compute) the form of the symmetry transformation and the conserved current in the (p, x) variables. This "shift symmetry" is often used as a diagnostic for the presence of Goldstone bosons. (c) [5] Next, suppose that the last symmetry of the Lagrangian is broken by a term V' = -61. Show that the Goldstone bosons also get a mass, and that my ~ e/ po where po is the vacuum expectation value of the p field. This behavior is seen in ferromagnets, in the presence of an external magnetic field, and in QCD, where the pion is the Goldstone boson and e represents the quark mass. (Pedantic people call this small mass particle a "pseudo Goldstone boson.") Oftentimes, the literature will make an assumption that one is working at momentum scales much less than the Higgs mass and the radial degree of free- dom is "frozen" to po